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Position of a Line W.R.T Hyperbola

Consider the ...

Question

Consider the hyperbola xy = 4 and a line 2x + y = 4. O is the centre of the hyperbola. Tangent at any point P on the hyperbola intersects the coordinate axes at A and B.

Locus of circumcentre of $Δ O A B$ is ___

An ellipse with $e = \frac{1}{\sqrt{2}}$

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A circle

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A parabola

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A hyperbola with $e = \sqrt{2}$

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Solution

The correct option is C
A parabola

Locus of circum centre is rectangular hyperbola.
R = (2, 0) Any point through R is $(2 + r c o s θ, r s i n θ)$
$\Rightarrow (2 + r c o s θ) (r s i n θ) = 4 \Rightarrow r^{2} s i n θ c o s θ + 2 r s i n θ = 4 ∴ r_{1} r_{2} = ∣ ∣ \frac{4}{s i n θ c o s θ} ∣ ∣ \geq 8$

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Consider the hyperbola xy = 4 and a line 2x + y = 4. O is the centre of the hyperbola. Tangent at any point P on the hyperbola intersects the coordinate axes at A and B. Locus of circumcentre of ΔOAB is ___

The correct option is C A parabola Locus of circum centre is rectangular hyperbola. R = (2, 0) Any point through R is (2+r cos θ, r sin θ) ⇒(2+r cos θ)(r sin θ)=4⇒r2sin θ cos θ+2r sin θ=4∴r1r2=∣∣4sin θ cos θ∣∣≥8

Consider the hyperbola xy = 4 and a line 2x + y = 4. O is the centre of the hyperbola. Tangent at any point P on the hyperbola intersects the coordinate axes at A and B.

Locus of circumcentre of $Δ O A B$ is ___